Defects and patterns in the Calculus of Variations

这项研究的目标是对物理学中出现的变分问题和相关的偏微分方程组(PDE)的解进行严格的数学分析。对于许多物理系统，可实现的配置是那些最小化(全局或局部)能量泛函的配置。我们将研究的泛函包括：在超导背景下引入的Ginzburg-Landau模型，但也是凝聚态物理和材料中许多现象的范例；向列型液晶的Landau-de Gennes模型；以及两嵌段共聚物的Ohta-Kawasaki能量。这些模型有许多基本特征，当模型中的一个或多个参数被发送到零或无穷大时，这些特征在奇异摄动极限中显现出来。在这些极限区域中，可以观察到解发展出几何奇点，例如涡旋、倾斜或磁区壁，而这些缺陷给出了系统最显著的特征。 总体目标是开发新的分析工具来研究奇异摄动的变分问题，阐明在实验和模拟中观察到的物理现象。重点将放在向量值函数的变分问题上，其中欧拉-拉格朗日方程将是非线性偏微分方程组。通常用于研究单个偏微分方程组的许多工具并没有扩展到系统，因此分析挑战是相当大的，但数学兴趣是巨大的。例如，在液晶的Landau-de Gennes理论中，状态是用Q张量场(对称的无迹矩阵值函数)来描述的，而奇点往往是由于本征值的交叉而产生的。这给识别和定位具有微妙能量特征的奇点(线和环缺陷，或位于边界表面的“Boojum”)带来了分析障碍。此外，嵌段共聚物的数学模型包含了基本的非局部相互作用，这为描述这些材料的PDE增加了额外的一层复杂性。 解决这些问题将需要发展新的方法来解决非线性和非局部偏微分方程组。我将把我自己的想法与来自非线性和几何分析的创新融合在一起，包括尖锐的能量界限(通过涡球结构)；单调性和ETA椭圆性方法(为调和映射开发)；分叉技术；伽玛收敛技术(给出表征奇点形状和相互作用的极限能量)；集中紧凑法(将复杂结构的极小化分解为多个组成部分)。所获得的分析结果将使我们对这些模型及其所描述的现象有一个更全面、更可靠的理解，同时也为分析、几何和物理之间的丰富相互作用提供了新的视角。

The object of this research proposal is the rigorous mathematical analysis of variational problems arising in physics, and of the solutions of the associated systems of partial differential equations (PDE). For many physical systems the realizable configurations are those which minimize (globally or locally) an energy functional. Among the functionals we will study are: Ginzburg-Landau models, introduced in the context of superconductivity, but also a paradigm for many phenomena in condensed matter physics and materials; the Landau-de Gennes model for nematic liquid crystals; and the Ohta-Kawasaki energy for di-block copolymers. These models have many fundamental features which reveal themselves in singular perturbation limits, when one or more parameters in the model are sent to zero or infinity. In these limiting regime the solutions are observed to develop geometrical singularities, such as vortices, disclinations, or domain walls, and these defects give the most salient features of the system. The overall goal is to develop new analytical tools to study singularly perturbed variational problems, to shed light on physical phenomena observed in experiments and simulations. The emphasis will be on variational problems for vector-valued functions, for which the Euler-Lagrange equations will be systems of nonlinear PDE. Many tools normally employed in studying a single PDE do not extend to systems, and so the analytical challenges are considerable but the mathematical interest is great. For instance, in the Landau-de Gennes theory of liquid crystals the states are described by Q-tensor fields (symmetric traceless matrix-valued functions,) and the singularities often occur because of eigenvalue crossing. This presents analytical hurdles in identifying and localizing singularities (line and ring defects, or "boojums" lying on boundary surfaces) which have subtle energy signatures. Further, mathematical models of block copolymers incorporate essential nonlocal interactions, which adds an extra layer of complexity to the PDEs describing these materials. Solving these problems will entail the development of new methods for systems of nonlinear and nonlocal PDE. I will blend my own ideas with innovations coming from nonlinear and geometric analysis, including sharp energy bounds (via vortex-ball constructions); monotonicity and eta-ellipticity methods (developed for harmonic maps); bifurcation techniques; Gamma-convergence techniques (giving limiting energies characterizing the shape and interactions of singularities); and concentration-compactness methods (which break down minimizers with complex structure into component pieces.) The analytical results obtained will give a more complete and reliable understanding of these models and the phenomena they describe, while providing new perspectives on the rich interplay between analysis, geometry, and physics.